Question

Find the total charge of a 2D disc sitting in the x−y plane, centered at the origin, with a radius of 50 cm. The surface charge density of the disc is given by rho s ​ =rho 0 ​ r where rho 0 ​ =15μC/(m 3 )

Answer 1

The total charge of a 2D disc centered at the origin with a radius of 50 cm and a given surface charge density can be found by integrating the surface charge density over the entire area of the disc. In this case, the total charge is 0.46875 μC.

Step-by-step explanation:

The total charge of a 2D disc can be found by integrating the surface charge density over the entire area of the disc. In this case, the surface charge density is given by ρs = ρ0r, where ρ0 = 15μC/(m3) is a constant and r is the distance from the center of the disc.

The area of the disc is given by A = πr2 = π(0.5m)2 = π*0.25m2.

Therefore, the total charge Q can be found by integrating the surface charge density over the area of the disc:

Q = ∫ρs dA = ∫(ρ0r) dA = ∫(ρ0r) πr2 dr = πρ0 ∫r3 dr = πρ0(r4/4)|0^(0.5m) = (1/4)πρ0(0.54) = 0.03125πρ0 = 0.03125(15)μC = 0.46875μC.